A follow up to this exercise.

Using 1/16" PVC for the ends and schedule 40 1/2" PVC
for the core, I wound 2000 turns of #28 wire on the
form and put things together to test them.

It works with 240 VAC across the bridge rectifier now.

And yes, the coil heats up. It's trying to dissipate
1440 watts. (240 VAC @ 6 amps)

The contacts on the transfer switch disconnect the
solenoid coil from power as soon as it starts to move.
Inertia carries it through the sequence.

Jeff-1.0